Page:Zur Thermodynamik bewegter Systeme.djvu/14

Namely, it is according to (8):

${\frac {\partial T}{\partial \beta }}=T_{0}{\frac {\partial ^{2}H}{\partial \beta \ \partial U_{0}}}=-{\frac {1}{1-\beta ^{2}}}T_{0}\beta {\frac {\partial }{\partial U_{0}}}(H+pv)=$ $=-{\frac {1}{1-\beta ^{2}}}\beta T_{0}{\frac {\partial H}{\partial U_{0}}}-{\frac {1}{1-\beta ^{2}}}\beta vT_{0}{\frac {\partial p}{\partial U_{0}}}.$

However, it is now

$T_{0}{\frac {\partial p}{\partial U_{0}}}=T_{0}\left({\frac {\partial p_{0}}{\partial U_{0}}}{\frac {\partial H}{\partial U_{0}}}+p_{0}{\frac {\partial ^{2}H}{\partial U_{0}^{2}}}-{\frac {\partial ^{2}H}{\partial U_{0}\partial v}}\right).$

If we insert herein

$T_{0}{\frac {\partial p_{0}}{\partial U_{0}}}=p_{0}{\frac {\partial T_{0}}{\partial U_{0}}}-{\frac {\partial T_{0}}{\partial v}},$

a relation known as following from the thermodynamics of resting bodies, then it becomes:

$T_{0}{\frac {\partial p}{\partial U_{0}}}=\left(p_{0}{\frac {\partial T_{0}}{\partial U_{0}}}-{\frac {\partial T_{0}}{\partial v}}\right)\cdot {\frac {\partial H}{\partial U_{0}}}+p_{0}T_{0}{\frac {\partial ^{2}H}{\partial U_{0}^{2}}}-T_{0}{\frac {\partial ^{2}H}{\partial U_{0}\partial v}}=$ $=p_{0}{\frac {\partial }{\partial U_{0}}}\left(T_{0}{\frac {\partial H}{\partial U_{0}}}\right)-{\frac {\partial }{\partial v}}\left(T_{0}{\frac {\partial H}{\partial U_{0}}}\right)=p_{0}{\frac {\partial T}{\partial U_{0}}}-{\frac {\partial T}{\partial v}}.$

Thus

${\frac {\partial T}{\partial \beta }}=-{\frac {\beta }{1-\beta ^{2}}}\cdot T-{\frac {\beta }{1-\beta ^{2}}}v\left(p_{0}{\frac {\partial T}{\partial U_{0}}}-{\frac {\partial T}{\partial v}}\right)$

and

$dT=-{\frac {\beta d\beta }{1-\beta ^{2}}}T.$

The change of temperature is thus the same for all bodies.

This expression as well as the expression for $dv$ (12) can be immediately integrated. Then we obtain:

$v=v_{0}{\sqrt {1-\beta ^{2}}},$ $T=T_{0}{\sqrt {1-\beta ^{2}}}.$

(14)